(1)
In Theorem REF , the presence of zero resonances has no effect on the \(p\) -range of the \(L^p\) -boundedness of wave operators \(W_\pm , W_\pm ^*\) , and only require that the potentials \(V\)  satisfy stronger decay conditions than the regular case.

(2)
We in fact prove the following bounds with an explicit dependence on the weights:
\({\Vert W_\pm f\Vert }_{L^p(w_p)}&\lesssim [w_p]_{A_p}^{\max \lbrace 1,1/(p-1)\rbrace }({\Vert f\Vert }_{L^p(w_p)}+{\Vert \tau f\Vert }_{L^p(w_p)}),\quad 1<p<\infty ,\\{\Vert W_\pm f\Vert }_{L^{1,\infty }(w_1)}&\lesssim [w_1]_{A_1}(1+\log [w]_{A_1})({\Vert f\Vert }_{L^1(w_1)}+{\Vert \tau f\Vert }_{L^1(w_1)}),\) 

where \([w]_{A_p}\)  is the \(A_p\) -characteristic constant of \(w\)  (see Appendix ) and the implicit constants are independent of \(w_p,w_1\) . Moreover, the same bounds also hold for \(W_\pm ^*\) . The estimates of type [theorem11]theorem11 (without \({\Vert \tau f\Vert }_{L^p(w_p)}\) ) are known as the \(A_p\) -estimates in the theory of Calderón–Zygmund operators and known to be sharp (see [1]}). We also refer to [2]} for the estimates of type [theorem12]theorem12 for Calderón–Zygmund operators.

(3)
For the Schrödinger operator \(-\Delta +V(x)\)  on \({\mathbb {R}}^3\) , Beceanu [3]} proved a weighted \(L^p\) -boundedness of the wave operators with a specific weight \({\langle }x{\rangle }^a\)  for \(|a|<1\)  under a suitable assumption on \(V\)  depending on \(a\) . Compared with his result, the interesting point of Theorem REF  (2) is that we can take general even (i.e. radial) weight \(w_p\in A_p\) . Moreover, our assumption on \(V\)  is independent of the choice of weights.


